#10440031

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,12)(3,8,14,16)(4,5,10,9)(6,15,13,11)(17,18,23,26)(19,24,21,29)(20,22,25,32)(27,28,30,31)(34,39)(36,43)(41,45)(44,48);(1,3,7,14)(2,8,12,16)(4,13,10,6)(5,11,9,15)(19,21)(24,29)(27,30)(28,31)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,46)(47,48);(1,4)(2,9)(3,13)(5,12)(6,14)(7,10)(8,15)(11,16)(17,19,23,21)(18,24,26,29)(20,27,25,30)(22,28,32,31)(33,35,38,40)(34,36,39,43)(37,42,46,47)(41,44,45,48);(1,4)(2,5)(3,13)(6,14)(7,10)(8,11)(9,12)(15,16)(17,20)(18,22)(19,27)(21,30)(23,25)(24,28)(26,32)(29,31)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,48)(44,47);(1,5)(2,10)(3,11)(4,12)(6,8)(7,9)(13,16)(14,15)(17,21,23,19)(18,24,26,29)(20,27,25,30)(22,31,32,28)(33,36)(34,40)(35,39)(37,44)(38,43)(41,47)(42,45)(46,48);(1,6,7,13)(2,11,12,15)(3,4,14,10)(5,8,9,16)(17,22)(18,25)(19,28)(20,26)(21,31)(23,32)(24,30)(27,29)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10347227);